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arXiv:2405.00209 · Physical Review Research

Superluminal matter waves

John P. Palastro, D. Ramsey, Martin S. Formanek, J. Vieira, ANTONINO DI PIAZZA

Published Oct 29, 2024• 4 citations
arXivHTMLPDF

Abstract

The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.043075

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The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values. Published by the American Physical Society 2024
For Klein?Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability Psuper, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. Psuper?depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein?Gordon waves and 1?1/ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for Psuper.
Dirac sought a relativistic quantum free-particle Hamiltonian that imposes space-time symmetry on the Schroedinger equation in configuration representation; he ignored the Lorentz covariance of energy-momentum. Dirac free-particle velocity therefore is momentum-independent, breaching relativity basics. Dirac also made solutions of his equation satisfy the Klein-Gordon equation via requirements imposed on its operators. Dirac particle speed is thereby fixed to the unphysical value of c times the square root of three, and anticommutation requirements prevent four observables, including the components of velocity, from commuting when Planck's constant vanishes, a correspondence-principle breach responsible for Dirac free-particle spontaneous acceleration (zitterbewegung) that diverges in the classical limit. Nonrelativistic Pauli theory contrariwise is physically sensible, and its particle rest-frame action can be extended to become Lorentz invariant. The consequent Lagrangian yields the corresponding closed-form relativistic Hamiltonian when magnetic field is absent, otherwise a successive-approximation regime applies.
Credible reasons are presented to reveal that many of the lingering century old enigmas, surrounding the behavior of at least an individual quantum particle, can be comprehended in terms of an objectively real specific wave function. This wave function is gleaned from the single particle energy-momentum eigenstate offered by the theory of space filling universal quantum fields that is an inevitable outcome of Dirac's pioneering masterpiece. Examples of these well-known enigmas are wave particle duality, the de Broglie hypothesis, the uncertainty principle, wave function collapse, and predictions of measurement outcomes in terms of probability instead of certainty. Paul Dirac successfully incorporated special theory of relativity into quantum mechanics for the first time. This was accomplished through his ingenious use of matrices that allowed the equations of motion to maintain the necessary first order time derivative feature necessary for positive probability density. The ensuing Dirac equation for the electron led to the recognition of the mystifying quantized spin and magnetic moment as intrinsic properties in contrast to earlier ad hoc assumptions. The solution of his relativistic equation for the hydrogen atom produced results in perfect agreement with experimental data available at the time. The most far reaching prediction of the celebrated Dirac equation was the totally unexpected existence of anti-particles, culminating in the eventual development of the quantum field theory of the Standard Model that reveals the deepest secrets of the universe known to date. Quanta 2019; 8: 88–100.
Solitary-particle quantum mechanics' inherent compatibility with special relativity is implicit in Schroedinger's postulated wave-function rule for the operator quantization of the particle's canonical three-momentum, taken together with his famed time-dependent wave-function equation that analogously treats the operator quantization of its Hamiltonian. The resulting formally four-vector equation system assures proper relativistic covariance for any solitary-particle Hamiltonian operator which, together with its canonical three-momentum operator, is a Lorentz-covariant four-vector operator. This, of course, is always the case for the quantization of the Hamiltonian of a properly relativistic classical theory, so the strong correspondence principle definitely remains valid in the relativistic domain. Klein-Gordon theory impairs this four-vector equation by iterating and contracting it, thereby injecting extraneous negative-energy solutions that are not orthogonal to their positive-energy counterparts of the same momentum, thus destroying the basis of the quantum probability interpretation. Klein-Gordon theory, which thus depends on the square of the Hamiltonian operator, is as well thereby cut adrift from Heisenberg's equations of motion. Dirac theory confuses the space-time symmetry of the four-vector equation system with such symmetry for its time component alone, which it fatuously imposes, thereby breaching the strong correspondence principle for the free particle and imposing the starkly unphysical momentum-independence of velocity. Physically sensible alternatives, with external electromagnetic fields, to the Klein-Gordon and Dirac equations are derived, and the simple, elegant symmetry-based approach to antiparticles is pointed out.
A single-particle Hamiltonian independent of the particle's coordinate ensures the conserves momentum, i.e., is free. Lorentz-covariance of that Hamiltonian's energy-momentum specifies it up to the particle's rest energy; the it describes has speed below c and constant velocity parallel to its conserved momentum. Dirac took his free-particle Hamiltonian to have the same squared value as that relativistic one, but unwittingly blocked Lorentz-covariance of his Hamiltonian's energy-momentum by requiring it to be inhomogeneously linear in momentum. The Dirac free particle badly flouts relativity and even physical cogency; its velocity direction is extremely nonconstant, while its speed is fixed to c times the square root of three even when it interacts electromagnetically. Both its rest energy and total energy can be negative, and its velocity components and rest energy are artificially correlated by being mutually anticommuting; its alleged spin is an artifact of the anticommutation of its velocity components. Unlike the Dirac Hamiltonian, the nonrelativistic Pauli Hamiltonian is apparently physically sensible for speed far below c. Its relativistic extension is worked out via Lorentz-invariant upgrade of its associated action functional at zero velocity, and is obtained in closed form if there is no applied magnetic field; a successive approximation scheme must otherwise be used.